PART I. Short-answer questions. [35 minutes] (40 POINTS)

1.   For some value z0 , the probability that a standard normal variable is between z0   and

0  is 0.4798 (or P( z0 < Z <0)). What is the value of z0 ? Show work below.

2.   Find the value of F01,9,18  . Show your work below.

3.   Let X be a binomial random variable with n=60 and p = .04. Use the appropriate Poisson distribution to approximate P(X=7). Show work below.

4.   "A continuous random variable is one that can have only certain clearly separated values resulting from a count of some item of interest."

Circle one:     True / False

5.   A minister has to work every Saturday on which there is a wedding. Church records show  that  the  number  of weddings  that  take  place  on  Saturdays  has  a  Poisson distribution with mean 1. What is the probability that the minister will have to work this Saturday? Show work below

6.   The probability distribution of a continuous random variable is represented by a that is a smooth curve.

7.   Consider a random variable X with the following probability distribution.

x                      -4        0          1          2

p(x)                 .2         .3         .4         .1

Find P(0  X  1). Show work below

8.   Toss three fair coins and let X equal the number of heads observed. Find P(X > 2).

Show work below

9.   Find P(t27  > 1.90) . Show your work below.

10. The standard deviation of a sampling distribution is called                                            

11. “If the  random  variable X is  exponentially  distributed  and  the  parameter  of the distribution     = 4, then P(X ³ 1) = 0.25.”

Circle one:     True / False / Uncertain

12. The  time  required  to  complete  a  particular  assembly  operation  has  a  uniform distribution between 25 and 50 minutes. What is the probability that the assembly operation will require more than 40 minutes to complete? Show your work below.

13. A portfolio is composed of two stocks. The proportion of each stock, their expected values, and standard deviations are listed:

Stock                                            1                      2

Proportion of Portfolio              .30                   .70

Mean                                            .12                   .25

Standard Deviation                    .02                   .15

Calculate the expected value and standard deviation of the portfolio. Assume that p= .2 .

Show your work below.

14. Find P(X236  > 25) . Show your work below.

15. What properties does the normal distribution have?

Circle:            unimodal / symmetric / mean=median=mode / All of the above

16. Suppose X is a normally distributed random variable with mean 120 and variance 36. Find x0 corresponding to the 80th percentile. Show work below.

17. A population (with unknown distribution) has a mean of   = 40 and a standard deviation of 12. What does the central limit theorem say about the sampling distribution of the mean if samples of size 100 are drawn from this population?                              

18. The computer chips in today's notebook and laptop computers are produced from       semiconductor wafers. Certain semiconductor wafers are exposed to an environment that generates up to 100 possible defects per wafer. The number of defects per wafer, X, was found to follow a binomial distribution if the manufacturing process is stable  and generates defects that are randomly distributed on the wafers (IEEE Transactions on Semiconductor Manufacturing, May 1995). Let p represent the probability that a   defect occurs at any one of the 100 points of the wafer. If p=.50, determine whether   the normal approximation can be used to characterize X. Show work below.

19. Find P(F12,16  >2.89). Show your work below.

20. Provide two objectives of the sampling distribution of statistics:

EXTRA CREDIT:

Describe the sampling distribution of X

PART II. Problems (60 POINTS)

1.    Problem 1 (20 points)

Due to an increasing number of non-performing loans, a Texas bank now insists that   several stringent conditions be met before a customer is granted a consumer loan. As a result, 60% of all customers applying for a loan are rejected.

a.   Suppose 15 new loan applications are selected at random.

(i)         Find the expected value (mean) and the standard deviation of the number of loan applications that are accepted.

(ii)       What is the probability that at least 10 are accepted? (Show all your work).

(iii)      What is the probability that no more than 6 are accepted? (Show all your work).

b.   Suppose now that 50 new loan applications are selected at random, but 98% of all customers applying for a loan are rejected..

(i)        Find the expected value (mean) and the standard deviation of the number of loan applications that are accepted.

(ii)       What is the probability that more than 25 are accepted? (Show all your work).

2.   Problem 2 (20 points)

Cattle are often fattened in a feedlot before being shipped to a slaughterhouse. Suppose the weight gain per steer at a feedlot average 1.75 pounds per day, with a standard deviation of 0.30 pounds.  Assume that the weight gain is normal distributed.

a.   What is the probability a steer will gain over two pounds on a given day? (Show your work)

b.   Determine the probability a steer will gain between 2 and 3 pounds in a given day. (Show your work).

c.   Provide the probability of selecting 3 steers that gain less than 1.5 pounds on a given day, assuming the 3 weight gains are independent. [Hint: use the multiplication rule with independent events]. (Show your work).

d.   Compute the probability that the weight gain is within one standard deviation of the mean? (Show your work).

e.   Compute the probability that the weight gain is within three standard deviations of the mean? (Show your work).

3.   Problem 3 (20 points)

The amount of time spent by North American adults watching television per day is normally distributed with a mean of 8 hours and a standard deviation of 2 hours.

a.   What proportion of the population watches television for more than 9 hours per day? Round your answer to the nearest 100th decimal points. (Show your work)

b.   What is the probability that the average number of hours spent watching television by a random sample of 10 adults is more than 9 hours? (Show your work)

c.   What is the probability that the average number of hours spent watching television by a random sample of 25 adults is between 6 hours and 10 hours? (Show your work)

d.   What is the probability that in a random sample of 10 adults, more than 6 adults watch television for more than 9 hours per day? (Show your work)